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Difference between revisions of "2008 AIME I Problems/Problem 15"

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__TOC__
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== Problem ==
 
== Problem ==
−
A square piece of paper has sides of length <math>100</math>. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance <math>\sqrt{17}</math> from the corner, and they meet on the diagonal at an angle of <math>60^{\circ}</math> (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upped edges, can be written in the form <math>\sqrt[n]{m}</math>, where <math>m</math> and <math>n</math> are positive integers, <math>m<1000</math>, and <math>m</math> is not divisible by the <math>n</math>th power of any prime. Find <math>m+n</math>.
+
A square piece of paper has sides of length <math>100</math>. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance <math>\sqrt{17}</math> from the corner, and they meet on the diagonal at an angle of <math>60^{\circ}</math> (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form <math>\sqrt[n]{m}</math>, where <math>m</math> and <math>n</math> are positive integers, <math>m<1000</math>, and <math>m</math> is not divisible by the <math>n</math>th power of any prime. Find <math>m+n</math>.
 
<center><asy>import cse5;
 
<center><asy>import cse5;
 
size(200);
 
size(200);
Line 19: Line 21:
 
MP("\mathrm{fold}",(s+r/2*dir(45).x,r*dir(45).y));</asy></center>
 
MP("\mathrm{fold}",(s+r/2*dir(45).x,r*dir(45).y));</asy></center>
  
−
__TOC__
+
== Picture for Solutions ==
−
== Solution ==
+
(Used for the following solutions)
 +
 
 
<center><asy>
 
<center><asy>
−
import three; import math;
+
import three; import math; import cse5; import olympiad;
 
size(500);
 
size(500);
−
pathpen=black;
+
pathpen=blue;
 
real r = (51^0.5-17^0.5)/200, h=867^0.25/100;
 
real r = (51^0.5-17^0.5)/200, h=867^0.25/100;
 
triple A=(0,0,0),B=(1,0,0),C=(1,1,0),D=(0,1,0);
 
triple A=(0,0,0),B=(1,0,0),C=(1,1,0),D=(0,1,0);
 
triple F=B+(r,-r,h),G=(1,-r,h),H=(1+r,0,h),I=B+(0,0,h);
 
triple F=B+(r,-r,h),G=(1,-r,h),H=(1+r,0,h),I=B+(0,0,h);
−
D(B--F--H--cycle); D(B--F--G--cycle);
+
draw(B--F--H--cycle); draw(B--F--G--cycle);
−
D(G--I--H); D(B--I); D(A--B--C--D--cycle);
+
draw(G--I--H); draw(B--I); draw(A--B--C--D--cycle);
 
triple Fa=A+(-r,-r, h), Fc=C+(r,r, h), Fd=D+(-r,r, h);
 
triple Fa=A+(-r,-r, h), Fc=C+(r,r, h), Fd=D+(-r,r, h);
 
triple Ia = A+(0,0,h), Ic = C+(0,0,h), Id = D+(0,0,h);
 
triple Ia = A+(0,0,h), Ic = C+(0,0,h), Id = D+(0,0,h);
−
D(Ia--I--Ic); D(Fa--F--Fc--Fd--cycle);
+
draw(Ia--I--Ic); draw(Fa--F--Fc--Fd--cycle);
−
D(A--Fa); D(C--Fc); D(D--Fd);
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draw(A--Fa); draw(C--Fc); draw(D--Fd);
 +
triple A1 = (F.x,F.y,0); real factor = 1/4;
 +
draw(F--A1--B); draw((A1+(F-A1)*factor)--(A1+(F-A1)*factor+(B-A1)*factor)--(A1+(B-A1)*factor));  
 
</asy></center>
 
</asy></center>
−
=== Solution 1 ===
+
 
 +
== Solution 1 ==
 
In the original picture, let <math>P</math> be the corner, and <math>M</math> and <math>N</math> be the two points whose distance is <math>\sqrt{17}</math> from <math>P</math>. Also, let <math>R</math> be the point where the two cuts intersect.
 
In the original picture, let <math>P</math> be the corner, and <math>M</math> and <math>N</math> be the two points whose distance is <math>\sqrt{17}</math> from <math>P</math>. Also, let <math>R</math> be the point where the two cuts intersect.
  
 
Using <math>\triangle{MNP}</math> (a 45-45-90 triangle), <math>MN=MP\sqrt{2}\quad\Longrightarrow\quad MN=\sqrt{34}</math>. <math>\triangle{MNR}</math> is [[equilateral triangle|equilateral]], so <math>MR = NR = \sqrt{34}</math>. (Alternatively, we could find this by the [[Law of Sines]].)
 
Using <math>\triangle{MNP}</math> (a 45-45-90 triangle), <math>MN=MP\sqrt{2}\quad\Longrightarrow\quad MN=\sqrt{34}</math>. <math>\triangle{MNR}</math> is [[equilateral triangle|equilateral]], so <math>MR = NR = \sqrt{34}</math>. (Alternatively, we could find this by the [[Law of Sines]].)
  
−
The length of the perpendicular from <math>P</math> to <math>MN</math> in <math>\triangle{MNP}</math> is <math>\frac{\sqrt{17}}{\sqrt{2}}</math>, and the length of the perpendicular from <math>R</math> to <math>MN</math> in <math>\triangle{MNR}</math> is <math>\frac{\sqrt{51}}{\sqrt{2}}</math>. Adding those two lengths, <math>PR=\frac{\sqrt{17}+\sqrt{51}}{\sqrt{2}}</math>. (Alternatively, we could have used that <math>\tan 75^{\circ} = \tan (30+45) = \frac{\sqrt{6}+\sqrt{2}}{4}</math>.)
+
The length of the perpendicular from <math>P</math> to <math>MN</math> in <math>\triangle{MNP}</math> is <math>\frac{\sqrt{17}}{\sqrt{2}}</math>, and the length of the perpendicular from <math>R</math> to <math>MN</math> in <math>\triangle{MNR}</math> is <math>\frac{\sqrt{51}}{\sqrt{2}}</math>. Adding those two lengths, <math>PR=\frac{\sqrt{17}+\sqrt{51}}{\sqrt{2}}</math>. (Alternatively, we could have used that <math>\sin 75^{\circ} = \sin (30+45) = \frac{\sqrt{6}+\sqrt{2}}{4}</math>.)
  
 
Drop a [[perpendicular]] from <math>R</math> to the side of the square containing <math>M</math> and let the intersection be <math>G</math>.
 
Drop a [[perpendicular]] from <math>R</math> to the side of the square containing <math>M</math> and let the intersection be <math>G</math>.
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MP("P",P,SW); MP("N",N,SW); MP("M",M,SW); MP("R",R,NE);
 
MP("P",P,SW); MP("N",N,SW); MP("M",M,SW); MP("R",R,NE);
 
MP("G",G,SW);
 
MP("G",G,SW);
−
</asy><asy>import three; import math;
 
−
size(300);
 
−
pathpen=black;
 
−
real r=(51^0.5-17^0.5)/200, h=867^0.25/100;
 
−
triple A=(1,0,0), Ap=A+(r,-r,h), F=A+(0,0,h), N=(1,-r,h), M=(1+r,0,h);
 
−
D(A--Ap--M--cycle); D(A--Ap--N--cycle); D(N--F--M); D(A--F); D(F--Ap,dashed);
 
−
MP("\sqrt{34}",(A+Ap)/2,W);
 
−
MP("\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}",(F+N)/2,N);
 
−
MP("\frac{\sqrt{51}+\sqrt{17}}{2}",(A+7*M)/8,ENE);
 
−
MP("\sqrt[4]{867}",(F+A)/2,E);
 
−
MP("A",A,E);
 
−
MP("A'",Ap,W);
 
−
MP("F",F,E);
 
−
D(rightanglemark(A,F,M,0.05));
 
−
D(rightanglemark(Ap,N,A,0.05));
 
−
D(rightanglemark(Ap,M,F,0.05));
 
 
</asy></center>
 
</asy></center>
  
−
Let $ABCD$ be the smaller square base of the tray and let $A'B'C'D'$ be the larger square, such that $AA'$, etc, are edges. Let $F$ be the foot of the perpendicular from $A$ to plane $A'B'C'D'$.
+
Let <math>A'B'C'D'</math> be the smaller square base of the tray and let <math>ABCD</math> be the larger square, such that <math>AA'</math>, etc, are edges. Let <math>F</math> be the foot of the perpendicular from <math>A</math> to plane <math>A'B'C'D'</math>.
  
−
We know $AA'=MR=\sqrt{34}$ and $A'F=MG\sqrt{2}=\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}$. Now, use the Pythagorean Theorem on triangle $AFA'$ to find $AF$:
+
We know <math>AA'=MR=\sqrt{34}</math> and <math>A'F=MG\sqrt{2}=\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}</math>. Now, use the Pythagorean Theorem on triangle <math>AFA'</math> to find <math>AF</math>:
  
−
$$\begin{align*}\left(\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}\right)^2+AF^2&=\left(\sqrt{34}\right)^2<br />\frac{51-34\sqrt{3}+17}{2}+AF^2&=34<br />AF&=\sqrt{34-\frac{68-34\sqrt{3}}{2}}<br />AF&=\sqrt{\frac{34\sqrt{3}}{2}}<br />AF&=\sqrt[4]{867}\end{align*}$$
+
<cmath>\begin{align*}\left(\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}\right)^2+AF^2&=\left(\sqrt{34}\right)^2\\ \frac{51-34\sqrt{3}+17}{2}+AF^2&=34\\AF&=\sqrt{34-\frac{68-34\sqrt{3}}{2}}\\AF&=\sqrt{\frac{34\sqrt{3}}{2}}\\AF&=\sqrt[4]{867}\end{align*}</cmath>
  
−
The answer is $867 + 4 = \boxed{871}$.
+
The answer is <math>867 + 4 = \boxed{871}</math>.
  
−
=== Solution 2 ===
+
== Solution 2 ==
 
In the final pyramid, let <math>ABCD</math> be the smaller square and let <math>A'B'C'D'</math> be the larger square such that <math>AA'</math>, etc. are edges.
 
In the final pyramid, let <math>ABCD</math> be the smaller square and let <math>A'B'C'D'</math> be the larger square such that <math>AA'</math>, etc. are edges.
  
 
It is obvious from the diagram that <math>\angle A'AB = \angle A'AD = 105^\circ</math>.
 
It is obvious from the diagram that <math>\angle A'AB = \angle A'AD = 105^\circ</math>.
  
−
Let <math>AB</math> and <math>AD</math> be the positive <math>x</math> and <math>y</math> axes in a 3-d coordinate system such that <math>A'</math> has a positive <math>z</math> coordinate. Let <math>\alpha</math> be the angle made with the positive <math>x</math> axis. Define <math>\beta</math> and <math>\gamma</math> analogously.
+
Let <math>AB</math> and <math>AD</math> be the positive <math>x</math> and <math>y</math> axes in a 3-d coordinate system such that <math>A'</math> has a positive <math>z</math> coordinate. Let <math>\alpha</math> be the angle made with the positive <math>x</math> [[axis]]. Define <math>\beta</math> and <math>\gamma</math> analogously.
  
 
It is easy to see that if <math>P: = (x,y,z)</math>, then <math>x = AA'\cdot \cos\alpha</math>. Furthermore, this means that <math>\frac {x^2}{AA'^2} + \frac {y^2}{AA'^2} + \frac {z^2}{AA'^2} = \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1</math>.
 
It is easy to see that if <math>P: = (x,y,z)</math>, then <math>x = AA'\cdot \cos\alpha</math>. Furthermore, this means that <math>\frac {x^2}{AA'^2} + \frac {y^2}{AA'^2} + \frac {z^2}{AA'^2} = \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1</math>.
Line 104: Line 94:
 
Now, <math>z = AA'\cdot \cos\gamma = \sqrt [4]{34^2\cdot \frac {3}{4}} = \sqrt [4]{867}</math>.
 
Now, <math>z = AA'\cdot \cos\gamma = \sqrt [4]{34^2\cdot \frac {3}{4}} = \sqrt [4]{867}</math>.
  
−
It follows that the answer is <math>867 + 4 = \boxed{871}</math>.
+
It follows that the answer is <math>867 + 4 = \boxed{871}</math>.~Shen Kislay Kai
  
 
== See also ==
 
== See also ==

Latest revision as of 12:28, 3 September 2024

Problem

A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $\sqrt[n]{m}$, where $m$ and $n$ are positive integers, $m<1000$, and $m$ is not divisible by the $n$th power of any prime. Find $m+n$.

[asy]import cse5; size(200); pathpen=black; real s=sqrt(17); real r=(sqrt(51)+s)/sqrt(2); D((0,2*s)--(0,0)--(2*s,0)); D((0,s)--r*dir(45)--(s,0)); D((0,0)--r*dir(45)); D((r*dir(45).x,2*s)--r*dir(45)--(2*s,r*dir(45).y)); MP("30^\circ",r*dir(45)-(0.25,1),SW); MP("30^\circ",r*dir(45)-(1,0.5),SW); MP("\sqrt{17}",(0,s/2),W); MP("\sqrt{17}",(s/2,0),S); MP("\mathrm{cut}",((0,s)+r*dir(45))/2,N); MP("\mathrm{cut}",((s,0)+r*dir(45))/2,E); MP("\mathrm{fold}",(r*dir(45).x,s+r/2*dir(45).y),E); MP("\mathrm{fold}",(s+r/2*dir(45).x,r*dir(45).y));[/asy]

Picture for Solutions

(Used for the following solutions)

[asy] import three; import math; import cse5; import olympiad; size(500); pathpen=blue; real r = (51^0.5-17^0.5)/200, h=867^0.25/100; triple A=(0,0,0),B=(1,0,0),C=(1,1,0),D=(0,1,0); triple F=B+(r,-r,h),G=(1,-r,h),H=(1+r,0,h),I=B+(0,0,h); draw(B--F--H--cycle); draw(B--F--G--cycle); draw(G--I--H); draw(B--I); draw(A--B--C--D--cycle); triple Fa=A+(-r,-r, h), Fc=C+(r,r, h), Fd=D+(-r,r, h); triple Ia = A+(0,0,h), Ic = C+(0,0,h), Id = D+(0,0,h); draw(Ia--I--Ic); draw(Fa--F--Fc--Fd--cycle); draw(A--Fa); draw(C--Fc); draw(D--Fd); triple A1 = (F.x,F.y,0); real factor = 1/4; draw(F--A1--B); draw((A1+(F-A1)*factor)--(A1+(F-A1)*factor+(B-A1)*factor)--(A1+(B-A1)*factor)); [/asy]

Solution 1

In the original picture, let $P$ be the corner, and $M$ and $N$ be the two points whose distance is $\sqrt{17}$ from $P$. Also, let $R$ be the point where the two cuts intersect.

Using $\triangle{MNP}$ (a 45-45-90 triangle), $MN=MP\sqrt{2}\quad\Longrightarrow\quad MN=\sqrt{34}$. $\triangle{MNR}$ is equilateral, so $MR = NR = \sqrt{34}$. (Alternatively, we could find this by the Law of Sines.)

The length of the perpendicular from $P$ to $MN$ in $\triangle{MNP}$ is $\frac{\sqrt{17}}{\sqrt{2}}$, and the length of the perpendicular from $R$ to $MN$ in $\triangle{MNR}$ is $\frac{\sqrt{51}}{\sqrt{2}}$. Adding those two lengths, $PR=\frac{\sqrt{17}+\sqrt{51}}{\sqrt{2}}$. (Alternatively, we could have used that $\sin 75^{\circ} = \sin (30+45) = \frac{\sqrt{6}+\sqrt{2}}{4}$.)

Drop a perpendicular from $R$ to the side of the square containing $M$ and let the intersection be $G$.

\begin{align*}PG&=\frac{PR}{\sqrt{2}}=\frac{\sqrt{17}+\sqrt{51}}{2}\\ MG=PG-PM&=\frac{\sqrt{17}+\sqrt{51}}{2}-\sqrt{17}=\frac{\sqrt{51}-\sqrt{17}}{2}\end{align*}

[asy]import cse5; size(200); pathpen=black; real s=sqrt(17), r=(sqrt(51)+s)/(sqrt(2)); pair P=(0,0), N=(0,sqrt(17)), M=(sqrt(17),0), R=r*dir(45), G=((sqrt(51)+sqrt(17))/2,0); D(2*N--P--2*M); D(N--R--M); D(P--R); D((R.x,2*N.y)--R--(2*M.x,R.y)); MP("30^\circ",R-(0.25,1),SW); MP("30^\circ",R-(1,0.5),SW); MP("\sqrt{17}",N/2,W); MP("\sqrt{17}",M/2,S); D(N--M,dashed); D(G--R,dashed); MP("P",P,SW); MP("N",N,SW); MP("M",M,SW); MP("R",R,NE); MP("G",G,SW); [/asy]

Let $A'B'C'D'$ be the smaller square base of the tray and let $ABCD$ be the larger square, such that $AA'$, etc, are edges. Let $F$ be the foot of the perpendicular from $A$ to plane $A'B'C'D'$.

We know $AA'=MR=\sqrt{34}$ and $A'F=MG\sqrt{2}=\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}$. Now, use the Pythagorean Theorem on triangle $AFA'$ to find $AF$:

\begin{align*}\left(\frac{\sqrt{51}-\sqrt{17}}{\sqrt{2}}\right)^2+AF^2&=\left(\sqrt{34}\right)^2\\ \frac{51-34\sqrt{3}+17}{2}+AF^2&=34\\AF&=\sqrt{34-\frac{68-34\sqrt{3}}{2}}\\AF&=\sqrt{\frac{34\sqrt{3}}{2}}\\AF&=\sqrt[4]{867}\end{align*}

The answer is $867 + 4 = \boxed{871}$.

Solution 2

In the final pyramid, let $ABCD$ be the smaller square and let $A'B'C'D'$ be the larger square such that $AA'$, etc. are edges.

It is obvious from the diagram that $\angle A'AB = \angle A'AD = 105^\circ$.

Let $AB$ and $AD$ be the positive $x$ and $y$ axes in a 3-d coordinate system such that $A'$ has a positive $z$ coordinate. Let $\alpha$ be the angle made with the positive $x$ axis. Define $\beta$ and $\gamma$ analogously.

It is easy to see that if $P: = (x,y,z)$, then $x = AA'\cdot \cos\alpha$. Furthermore, this means that $\frac {x^2}{AA'^2} + \frac {y^2}{AA'^2} + \frac {z^2}{AA'^2} = \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

We have that $\alpha = \beta = 105^\circ$, so $\cos^2 105^\circ + \cos^2105^\circ + \cos^2\gamma = 1\implies \cos\gamma = \sqrt [4]{\frac {3}{4}}$.

It is easy to see from the Law of Sines that $\frac {AA'}{\sin 45^\circ} = \frac {\sqrt {17}}{\sin 30^\circ}\implies AA' = \sqrt {34}$.

Now, $z = AA'\cdot \cos\gamma = \sqrt [4]{34^2\cdot \frac {3}{4}} = \sqrt [4]{867}$.

It follows that the answer is $867 + 4 = \boxed{871}$.~Shen Kislay Kai

See also

2008 AIME I (Problems • Answer Key • Resources)
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